Cantor Confusion
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From: Virgil on 12 Dec 2006 15:06 In article <457ece72(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <457d8cc0$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > > writes: > > > Dik T. Winter wrote: > > > > In article <1165761763.908889.34550(a)80g2000cwy.googlegroups.com> > > > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > > > ... > > > > > Let P(a) be the probability that an arbitrary natural is divisible > > > > > by > > > > > a fixed natural a. Then P(a) = 1/a . Forbidden by set theory. > > > > > > > > No. Not specifically forbidden by set theory. Forbidden because there > > > > are > > > > no appropriate definitions for the words you are using (they are not > > > > used > > > > conforming to standard definitions, so you better supply definitions). > > > > In probability theory (as is commonly use) you have to define how you > > > > *select* your arbitrary natural. You have not done so, so probability > > > > theory does not have an answer. > > > > > > Why does that matter? > > > > It does matter because if you do not properly define your problem, > > mathematics is not able to give an answer. > > It's sufficiently defined if one assumes that there is a uniform > probability distribution. From which assumption, added to the others, one can deduce that 0 = 1, and all sorts of peculiar things. > > > > > > This is the same thing as your stupid ball and > > > vase trick. Why do you need to label anything, or know what you're > > > choosing from the infinite set? > > > > Because that is part of the problem setting. Giving that setten will > > allow mathematics to model the question and give an answer. > > > > That problem has a clear answer with or without the labels: the sum > diverges as f(n)=9n. The labels are confounding, not clarifying. What is confounding to TO is clairifying to anyone with the wits to understand it. The result depends on the labeling. Eliminating the labeling makes the result impossible to determine. > > > And it is bad to think that because for a sequence of sets holds that > > lim{n -> oo} |S_n| = k > > with some particular value of k, that also > > | lim{n -> oo} S_n | = k > > because the latter statement contains something that has not been > > defined in mathematics. > > I'm not sure what that statement is supposed to say. Can uoi give an > example? > > But even when we define it, it is not certain > > that it holds. Given the following (I think reasonable) definition: > > lim{n -> oo} S_n = S > So, what, S_n is supposed to be an initial segment of the sequence? > > if: > > (1) for every element a in S there is an n0 such that a is in each of > > the sets S_n with n > n0 > > (2) for every element a not in S there is an n0 such that a is not in > > each of the sets S_n with n > n0. > In (2), it sounds like a would not exist in ANY S_n if it's not in S. > > > So from some particular point an element either remains in the sets in > > the sequence or remains out of the sets. > > You mean, at some point you can tell whether a given element a is in S, > because if it were, it would be there by then? > > > > > With this definition (when we look at the rationals) we have that > > lim{n -> oo} [0, 1/n] = [0] > Okay that interval degenerates to 0.... > > > and so: > > lim{n -> oo} | [0, 1/n] | = aleph0 != 1 = | lim{n -> oo} [0, 1/n] | > > (I am talking standard mathematics here). > > Are the |'s supposed to denote set size? If so, how can you claim that > [0,0] contains aleph_0 elements? > > > > > So taking cardinality and limits can not be interchanged except in some > > particular cases. But that is not unprecedented in mathematics. > > limits and integrals can also not be interchanged except in particular > > cases. And so can the interchange is not in general passoble if one > > of the things you interchange is a limit. Even interchanging limits > > is not in general possible. Consider: > > lim{x -> oo} lim{y -> oo} (2x + 3y)/xy > > True, but is it relevant? Yup! TO claims certain changes in order of operations make no difference. Here it is shown that such changes in order of operations can and often do make significant differences. So that TO must PROVE that his changes of order don't make a difference before he can claim they don't.
From: Virgil on 12 Dec 2006 15:16 In article <457ecfca(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > If the number of levels in the tree is countable, > then every path in the tree is finite, and marked by a specific edge > where the value arises. Is that not true? If TO means by "level" of a node in a binary tree the number of branches between it and the root node, then to have every path finite requires that every path end in a leaf node ( a ode with no child nodes ). That requires the number of "levels" be finite, not merely countable. > In order to allow infinitely > long strings, you must have an uncountable number of levels in the tree, > in which case 1/3 will exist with a specific edge, infinitely far from > the root node. There are no such things as trees with uncountably many "levels", at least according to any standard definition of trees. there is a unique root node at level 0, and each level thereafter is the successor to a previous level through a connected chain of levels down to the root.
From: Franziska Neugebauer on 12 Dec 2006 16:39 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: [...] >> 1. You do not present a convincing definition of "number". (Most >> likely you have none). > > Definitions are abbreviations like the following: [too long, too old, too German; no definition at all] >> 2. You do not present a convincing definition of "numbers" and "sets" >> which are "not fixed" or "un-fixed". >> >> 3. You do again try to discuss issues of neuro sciences >> (representation of abstract entities in mind (or in the brain?)) in >> sci math. > > Of course, because math requires mind and brain. Mind and brain and representation of (abstract) entities therein is still off topic in sci.math. F. N. -- xyz
From: Franziska Neugebauer on 12 Dec 2006 17:16 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Dik T. Winter schrieb: >> [...] >> >> Again you have provided neither a definition of "number", nor of >> >> "grow". >> >> Are you unable to do so? In common parlance, but that is not >> >> mathematics. In mathematics functions can grow in relation to >> >> their argument, but not the entities they denote. >> > >> > Functions cannot grow, according to modern mathematics. >> >> Wrong. I have provided a definition: >> >> ,----[ <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] >> | Definition: A function f: A |-> B grows iff there exist a1 < a2 of >> | dom(f) and f(a1) < f(a2). We use the abbreviation "f grows" for of >> | "the function f grows". >> `---- > > The natural number n is a particular set of n elements. > If the number n can take a value n_1 and can take a value n_2 > with n_1 =/= n_2 then the number n can vary. How is that related to your sentence that "functions cannot grow"? >> > The expression >> > "variable" is merely a relict from ancient times when people knew >> > that the objects of mathematics do not exist in some nirvana but >> > have to be present in a mind where not everything can be present >> > simultaneously. >> >> How do you call "Textbaustein" in English? > > Sorry, I did not expect that you read every word of mine addressed to > other people. You may assume that most of the subscribers at least skim over the postings of the threads of interest. Hence your copy and paste maneuver will hardly go unnoticed. F. N. -- xyz
From: Dik T. Winter on 12 Dec 2006 20:32
In article <1165921386.811691.96050(a)n67g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > I did not state that nodes represent numbers, I stated that it can be > > shown that nodes represent numbers. > > Great. That is set theory from the finest! "It is not the case, but it > can be shown or proved to be the case. " So do you negate what I wrote? Apparently you think that "stating that it can be proven that nodes represent numbers" is the same as "stating that nodes represent numbers". You added the adjective "erronously" to it. What was the error? Was there an error in the proof? If so, what part of the proof was in error? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |